Quantum Manipulation of Multi-Photon States of Light

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Introduction

Since the early times of Quantum Mechanics it was clear that this new theory repre-sented a breaking-point with the past deterministic interpretation of the physics world. If, from one side, this theory was able to unify concepts that were considered antithet-ical before its introduction, e.g. the wave-particle duality of matter and light, on the other hand it destroyed many pillars of our classical interpretation of nature. The concept of measurement itself, which represents the main instrument for physicists to investigate the world, was completely revolutionized, going from a completely deter-ministic interpretation to a probabilistic one, intrinsic in Born's rule [1]. In this thesis, instead of dealing with dierent philosophical implications of Quantum Mechanics, I will present the opportunities it opens to develop new technologies. After countless observations, which have conrmed its validity, Quantum Mechanics has begun to en-ter our daily life. The laser, widely used in medicine and in many other elds, as well as the transistors that are the building blocks of smartphones and computers that we use daily, are just some examples of devices whose working principles can be explained only in terms of this theory. In recent years, it has been understood that the features of Quantum Mechanics can be used to overcome the limits imposed by the classical interpretation of nature, especially in the eld of metrology, computation and com-munication. On the other hand, some operations easily implemented with classical systems are prohibited in the quantum domain, such as measuring a system without perturbing it or perfect cloning of units of information. The possible application of the so-called quantum revolution in technology is indeed a highly debated point, which is why basic research in the eld of quantum mechanics is still necessary today.

Light is a very powerful tool to investigate the validity of the predictions of Quantum Mechanics. Thanks to devices and methods available today, scientists are now able to generate, manipulate and characterize the states of a light system at the quantum level, making quantum state engineering a promising eld of investigation. In this con-text, the study of phenomena predicted by Quantum Mechanics can be carried out in

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three steps: the preparation of a system in a particular initial state, mathematically described by a Hilbert space; its manipulation, for which the basic instruments are described in this thesis; and nally, its characterization. For the last point, the Homo-dyne Detection technique, described in Section 1.3.2, is a powerful tool. In the context of state manipulation, many works [2][3][4] have underlined the extremely interesting possibilities opened by the ability to experimentally deal with the fundamental oper-ations of annihilation (ˆa) and creation (ˆa†_{) of single quanta of light. Over the years,}

more and more sophisticated techniques to implement quantum operations based on the experimental realization of the annihilation and creation operators have been de-veloped. For example, by exploiting the concept of quantum superposition, it has been possible to experimentally test the commutation relations between these operators [5], which are at the origin of the quantum nature of light. Following the lines of this experiment, I present in Chapter 2 a technique able to emulate, on weak quantum states of light, the same transformation caused by a strong optical nonlinearity, the Kerr eect, which can not be obtained with the materials available today.

Any discussion about the revolutions introduced by Quantum Mechanics can not be concluded without talking of entanglement. This is one of the most controversial con-cepts introduced by this theory, on which the most brilliant minds of the last century have been debating for a long time. At the beginning of Chapter 3, a general review of this phenomenon is presented. Again, light is a perfect tool to investigate it. At the end of this thesis I will show how, by delocalizing the addition operation among dierent light systems, it is possible to generate entanglement among them even if they are initially in a macroscopic non-entangled state. This experiment represents a new tool to study, in the macroscopic domain, phenomena up to now conned in the microscopic regime. From this work we can also understand what are the limits of the available technologies to when we deal with quantum eects.

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Chapter 1 Brief Introduction to Experimental

Quantum Optics

The aim of this work is to study some fundamental aspects of Quantum Mechanics, and in this rst chapter I will provide a description of the main building blocks using an experimental perspective. I will introduce various formalisms need to describe quantum operators, quantum states and quantum measurements.

1.1 Quantum Operators

Quantum mechanics is based on the concept that two systems can exchange only dis-crete quantities of energy. For example, an excited atom can jump to a lower energy state yielding a quantum of energy to the surrounding environment. From the mathe-matical point of view, the destruction and the generation of energy quanta are described by the ˆa and the ˆa† _{operators. Their action on a number state (|ni)}1 _{is described by}

the relations: ˆ a |ni =√n |n − 1i ˆ a |0i = 0, (1.1) ˆ a†|ni =√n + 1 |n + 1i (1.2)

1_{The details of this kind of state are better explained in Section 1.2. Here it is sucient to know}

that these vectors represent the eigenvectors of the hamiltonian describing the electromagnetic eld quantized in vacuum [1].

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and they obey the commutation rule:

ˆa, ˆa†_{= 1} _(1.3)

The relation (1.2) tells us that the creation operator (ˆa†₎ _{acts on a system with no}

energy, the vacuum (|0i), creating a single quantum of energy in that system. ˆ

a†|0i = |1i

For the destruction operator (ˆa) something similar is true. Its action on a system with exactly n quanta of excitation is to remove just one of them, leaving the system with n − 1 energy quanta.

This PhD thesis is focused on the study of the quantum properties of light systems, therefore, I will refer to the operators ˆa† _{and ˆa as the creators and the annihilators of}

photons, the energy quanta of the electromagnetic eld.

The product ˆa†_ˆ_a _{is another important operator that will be widely used in this work.}

It is called number operator (ˆn) and it has a crucial role in the quantization of the electromagnetic eld. For example, the Hamiltonian that describes the energy of a quantized electromagnetic eld in vacuum can be written in the form[1]

ˆ

H_{f ree}e.m. _{= ~ω ˆ}n + 1

2, (1.4)

where the frequency ω

2π denes the oscillation frequency of the eld. The role of ˆn is to

count the number of quantized excitations, each of energy ~ω, of the electromagnetic eld i.e. the number of photons. Applying ˆn on an eigenvector of the hamiltonian (1.4) we have2_:

ˆ

n |ni = n |ni . (1.5)

Equation (1.5) tells us that this operator leaves the number state unchanged, giving us the number of photons (n) that characterize it. I will show in the next session that the knowledge of the photon number statistics of an optical state gives interesting information about it. From this point of view, it is important to notice that the operator ˆn is hermitian, so it can be measured [6]. In Section 1.3 I will describe two detectors sensitive to the photon number carried by a state of light.

Other observables, very useful to characterize the quantum properties of a light state,

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are the quadratures of the electric eld, dened as a combination of the annihilation and creation operators:

ˆ XθM =

ˆ

a e−iθM _{+ ˆ}_a†_eiθM

2 , (1.6)

where θM is the measurement phase. Its meaning will be claried in Section 1.3.2, where

I will give a detailed description of an instrument, the Homodyne Detector, capable to measure the quadrature values of a quantum state of light for all the possible measure-ment phases. Commonly, we call the quadrature measured at phase θM = 0 as the ˆX

quadrature, while setting the phase equal to π

2 we measure the ˆY quadrature. This

convention is related to the mathematical form used to describe the monochromatic electric eld in terms of measurable operators

ˆ E(θM) = 1 2ˆae −iθM ₊ 1 2aˆ † eiθM = ˆX cos(θM) + ˆY sin(θM) , (1.7)

where I used the convention q2~ω

ε0V = 1. In Section 1.4 I will describe an algorithm

capable to give a complete description of an optical state starting from the results of a set of quadrature measurements performed on it, for dierent settings of θM.

1.2 Quantum States

In Section 1.1 an important concept starts to emerge. In quantum optics the role of the operators is to perform an action onto a system. It doesn't matter if it is a measurement or some other operation that manipulates a certain system, the operators contain the "rules of the game": they specify the action, the results depending on the particular state the system is in. In this section I will present a brief description of the main optical states used in this work.

1.2.1 Number states

The number states (or Fock States) are dened as the eigenstates of the number operator ˆ

n, as anticipated in Equation (1.5). They can be obtained by repeated application of the creation operator on a system initially in the vacuum state.

|ni = √1 n!â † 1⊗ â † 2⊗ · · · ⊗ â † n|0i (1.8)

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As an immediate consequence of this denition, we have that there is no uncertainty on the number of photons in such a state, so:

∆ˆn2

= hn| ˆn2|ni − hn| ˆn |ni2 = 0 (1.9)

From Equation (1.4) it is also simple to understand their natural predisposition to describe xed energy states of the electromagnetic eld. Indeed, it is possible to prove that they are an orthonormal and complete base on which to describe the solutions of the Schrödinger equation for the electromagnetic eld [1].

hn|mi = δn,m → Orthonormality condition ∞

X

n=0

|ni hn| = ˆI → Completeness condition (1.10)

Despite their easy mathematical description, they are non-trivial to produce in the laboratory. A lot of eorts have been spent on their generation in the past decades. The main procedures to produce them involve quantum dots[7][8], cold atoms [9][10], molecules [11][12], Nitrogen Vacancy Centers in diamonds [13][14] and nonlinear optical processes, such as Parametric Down Conversion, that will be described later in this work.

Let's focus now on the properties of this kind of states in relation to the electromagnetic eld. Using the rst part of Equation (1.7) and a few other relations presented in the previous section, it is possible to derive two important properties of measurements of electric eld performed on a number state.

Their mean value is always zero

hn| ˆE(θM) |ni = 0, (1.11)

and their variance grows as the photon number increases ∆ ˆE(θM)

2

= hn| ˆE2(θM) |ni − hn| ˆE(θM) |ni2

= 1 2(n +

1 2).

(1.12) The mean value and the variance of the electric eld for an n-photon Fock state do not depend on the phase at which the measurement is performed. This is the reason of the failure of any attempt to describe them in an eective noise theory. The measurement technique that I will describe in Section 1.3.2 allows to realize electric eld measure-ments, therefore, in view of this, it is useful to show the probability distribution of

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such measurements once performed on a given state. For the Fock state case, we can calculate these quantities in terms of the measurable quadrature operators as:

on the quadrature base, and Hn(

√

2XθM)are the Hermite polynomials. The quadrature

probability distributions for zero, one, and two-photon states are reported in Figure 1.1. From these plots it is evident that if we perform repeated quadrature measure-ments on a Fock state, the outcome distribution will remain unchanged regardless the measurement's reference phase.

Figure 1.1: Probability distributions of the ˆXθM quadrature for |0i, |1i and |2i Fock

states, for dierent measurement phases θM. These probability distributions are

in-variant when changing θM.

Before describing another important class of states, let us focus our attention on a particular number state, the vacuum state (|0i). This is the minimum-energy state of the electromagnetic eld and, from Equation (1.4), we can see that this energy is larger than zero. In quantum optics this is explained by considering the presence, everywhere in the space, of a randomly uctuating electric eld [15]. We will see that, using an

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homodyne detector, it is possible to measure also this eld. Let us now look at what happens if we perform a simultaneous measurement of the two quadratures ˆXand ˆY on the vacuum state. Because of the phase invariance of the quadrature distribution, the two measurements have the same uncertainty. From Equation (1.12) we can calculate the variance of the joint measurement as:

(∆ ˆX)2(∆ ˆY )2 = 1

16 (1.14)

This is the minimum value allowed by the Heisenberg uncertainty principle, obtained by the standard procedure [16], considering that, for the commutator between ˆX and

ˆ

Y, stands [ ˆX, ˆY ] = i

2. (1.15)

States that satisfy Equation (1.14) are called minimum uncertainty states.

1.2.2 Coherent states

This important class of states, introduced by Glauber in 1963 [17], is dened as the eigenstates of the annihilation operator:

ˆ

a |αi = α |αi _{α ∈ C} α = |α|eiθco (1.16)

where the eighenvalue α is a complex number that denes the amplitude |α| and the phase θco of the state. They can be expressed in the number state base as:

|αi = e−12|α| 2 ∞ X n=0 αn √ n!|ni , (1.17)

and recalling the expression of |ni (Equation (1.8)) as: |αi = e−12|α| 2 ∞ X n=0 (αâ†)n n! |0i = e (αâ†₋1 2|α| 2₎ |0i = e(αâ†−α∗â)|0i = ˆD(α) |0i , (1.18)

where we have dened the Displacement Operator as3 _{D(α) = e}ˆ (αˆa†_−α∗_ˆ_a)

. A deeper understanding of the action represented by this operator can be obtained by studying

3_{In this equation we have used the relation e}Aˆ_eBˆ _{= e}A+ ˆˆ B+1

2[A; ˆˆB], that is valid in this case because

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the properties of the electric eld of a coherent state. For the mean value of ˆE(θM)we

have:

hα| Ê(θM) |αi = hα| âe−iθM + â†eiθM |αi

= |α| cos(θM),

(1.19) where θM is the phase of the measurement. The variance of this operator has an

interesting property too: hα| ∆ ˆE(θM)

2

|αi = 1

4. _(1.20)

From these relations we can say that, applying the displacement operator to the vacuum state of the electric eld, we will shift the mean value of the results of a quadrature measurement from zero to |α| cos(θM), maintaining its variance unchanged. This fact

classies the coherent states as minimum uncertainty states, as the vacuum state. Unlike the number states, two dierent coherent states are, in general, not orthogonal:

hα|βi = e−|β|2+|α|22 X n,m (α∗)m_βn √ m!n! hn|mi = e−|β|2+|α|22 X n (α∗β)n n! = e−|α−β|22 . (1.21)

They can be considered orthogonal only in the limit |α − β| → ∞.

Coherent states are very useful in quantum optics because they are the best approx-imation of the ideal light state generated by a well-stabilized laser. Indeed, they are easy to produce and they will be widely used in this work. Also in this case, the prob-ability distribution for an electric eld measurement is an important quantity to keep in mind: Pα(XθM) = r 2 πe −2[X_θM−|α|cos(θM)]2_. (1.22)

From Figure 1.2 the dependence of the quadrature distributions on the phase of the measurement performed to obtain it is evident.

It is also useful to recall the photon number properties of this kind of states:

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Figure 1.2: Probability distributions of the ˆXθM quadrature for two dierent values of

α and dierent measurement phases θM.

(∆ˆn)2 = hα| ˆn2|αi − hα| ˆn |αi2 = |α|2 = hˆni. (1.24) Equations (1.23) and (1.24) are the rst two moments of the photon number probability distribution for a coherent state, shown in Figure 1.3 for three values of α.

5 10 15 20 25 30n 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Pn |α| = 1 5 10 15 20 25 30n 0.05 0.10 0.15 0.20 Pn |α| = 2 5 10 15 20 25 30n 0.02 0.04 0.06 0.08 0.10 0.12 Pn |α| = 3

Figure 1.3: Photon number probability distributions for coherent states of dierent mean photon number (hˆni = |α|2_).

This is a peaked Poisson distribution for which the relative photon number (intensity) uctuations decrease for increasing α

∆I ¯ I ∝ ∆ˆn hˆni = 1 phni. (1.25)

This fact qualify the coherent states as "the most classical one among quantum states". Indeed, for a classical stable wave the intensity is a xed quantity with no uctuations, that is the limit case for coherent states of large α.

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1.2.3 Pure States and Mixed States - The Density Operator

Formalism

All classes of states described so far are called pure states. In general, a pure state can be written as the superposition of states that form a base of the electromagnetic radiation eld ({|ψni}).

|ψi = |pure statei =X

n

cn|ψni , (1.26)

where cn are complex amplitudes that dene the probability to observe the state |ψni

(pn = |cn|2), and also the relative phase of the component |ψni respect to the overall

phase of the superposition. However, not all the states of the electromagnetic eld can be written as pure states. Imperfections in the generation apparatus, interactions with the external environment, or the will of the experimenter lead to a description of the light state only in terms of probability, where all the phase relations between the states are lost. This concept is introduced in quantum mechanics via the denition of the density operator (ˆρ) and these states are called statistical mixtures [1]

ˆ

ρ =X

n

pn|ψni hψn| . (1.27)

Here pn have a clear interpretation in term of probabilities, so we have:

X

n

pn = 1 (1.28)

From the relation (1.28) we can see that, as a special case of this formalism, we can also describe the pure states. Indeed, if only one of the pn (p∗) is dierent from zero,

we have p∗ = 1, so the resulting density matrix is

ˆ

ρ∗ = |ψ∗i hψ∗| , (1.29)

where |ψ∗i is a pure state of the base dened at the beginning of this section.

Describing the state of a system in terms of its density operator reveals its importance when we try to formalize the concept of expectation value of a measurement. In general, it is dened as the statistical mean of all the possible measurement outcomes. Let's say, for example, that the measurement is represented by the operator ˆO. The expectation value is:

h ˆOi =X

n

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but if we introduce the generic complete base {|aii}4 h Ôi =X n,i pnhψn| Ô |aii hai|ψni = =X n,i pnhai|ψni hψn| Ô |aii = X i hai| ˆρ Ô |aii = = T rρ ˆÔ, (1.31)

that is equivalent to the Equation (1.30) but easier to manipulate in the case of a statistical mixture. If we calculate the expectation value of the identity operator (ˆI), we nd the normalization condition of the density operator:

T r ( ˆρ) =X n,i pnhai|ψnihψn|aii = X n,i pnhψn|aiihai|ψni = X n pn= 1. (1.32)

If this condition is not veried for a density operator it is not representing a physical system.

It is also important to notice that, recalling the form of the density operator in the case of a pure states (Equation (1.29)), we have:

T r ˆρ2_pure = T r (ˆρpure) = 1. (1.33)

So T r(ˆρ2₎can be dened as the purity of a state and used to discern a pure state from

a statistical mixture. During this introduction to the density operator we made use of the base {|ψni}, over which ˆρ is diagonal5, but, in general, it is not an orthonormal

4P

i|aii hai| = ˆI

5_{Using this base indeed we have:}

hψn| ˆρ |ψni = pn ∀n

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base. Switching to such a base ({|aii}) the ˆρ operator has the form: ˆ ρ =X n pn|ψni hψn| = X n,i,j pn|aii hai|ψnihψn|aji haj| = X i,j ρij|aii haj| ρij = hai| ˆρ |aji = X n,i,j pnhai|ψnihψn|aji, (1.35)

where the diagonal elements (ρii =

P

npn|hai|ψni|2) again tell us the probability to

nd the system in the base state |aii. Instead, the o-diagonal elements are related to

the correlations between the base states for the physical system represented by ˆρ. In conclusion, we can say that the density operator contains all the information about the state it represents. In the last section of this chapter I will show a technique to obtain the density matrix ρij from a set of experimental measurements.

1.2.4 Distributed Modes - A more realistic formalism

The theory presented in the previous sections is the easiest way to explain the concepts of operator and state from the quantum optics point of view, but, in this form, it is often far from the real experimental situation. In several experiments the properties of the optical states are not simply described as in Section 1.2. Indeed, if we say that we have generated a single photon state, we are giving only a partial information. It could be very important to specify also the spectral band over which we have produced it, for example. Other important characteristics are its propagation direction, its temporal or spatial shape, polarization, and so on, depending on the type of experiment performed. All these features dene the mode of the optical state. From a practical point of view it is useful to split the concept of optical mode into subsets, each of which refers to dierent features. In this thesis I will use the term "spatial mode" to indicate the spatial properties of an optical state, as well as "spectral mode", "temporal mode", etc. To mathematically formalize this concept we have to slightly modify some of the above denitions. The operator ˆa†_{dened in Equation (1.2), for example, adds a single}

photon at a monochromatic frequency, with a single wave vector and so on. A more complete description of this operator is6_:

ˆ a†(ω) =

Z dω0

δ(ω − ω0)ˆa†(ω0), (1.36)

where it is explicit that only one monochromatic frequency mode is involved. From this expression it is easy to generalize to an operator that generates single photons with

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a wave-packet distribution of frequencies, as is very common for most of the devices used to produce them

ˆ a†_g =

Z dωg∗

(ω)ˆa†(ω). (1.37)

Here the notation is heavier than the one in Equation (1.2) but has to be interpreted as follows: ˆa†_(ω) _{is the monochromatic creation operator already dened, ˆa}†

g is the

operator that generates a single photon in a frequency wave-packet of shape dened by the complex mode function g(ω).

This formalism can alternatively be incorporated into the states instead of into the operators. For example, for the single photon state we will write

|1gi = ˆa†g|0i = Z dωg∗ (ω)ˆa†(ω) |0i = Z dωg∗ (ω) |1ωi , (1.38)

where the notation has the same meaning of Equation (1.37). This is useful for the calculations in which we don't want to introduce operators, but the mode properties have to be taken into consideration. For example, using this formalism, we can see that if two Fock states have equal number of photons, they can have a scalar product equal to zero if their modes (indicated here as spectral modes g(ω) and f(ω)) are not matched to each other, in contrast with the denition (1.10).

h1g|1fi = Z dω h0| g(ω)ˆa(ω)Z dω0_f∗_(ω0_)ˆ_a†_(ω0_{) |0i} = Z dωdω0 g(ω)f∗(ω0)h1ω|10ωi = Z dωdω0 g(ω)f∗(ω0)δ(ω − ω0) = Z dωg(ω)f∗ (ω). (1.39)

This simple calculation is useful to understand the idea of the mode overlap, that will be further discussed in the context of homodyne detection.

The mode overlap can be used to take into account the frequency mismatch between two optical states, the spatial misalignment that frequently occurs while setting up an experiment, and other imperfections. In the rest of this work I will use the following convention for the normalization of the mode function g(ω):

Z

dω|g(ω)|2 _{= 1,} _(1.40)

in order to keep valid the commutation rule (1.3), as well as all the other relations regarding the number states.

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We can use this formalism also for the coherent state case. The only change we have to do is regarding the normalization of the mode prole:

Z

dω|α(ω)|2 _{= hˆ}_ni, _(1.41)

where α(ω) is the function describing the spectral mode occupied by the coherent state and hˆni is its mean photon number. With this convention, the denition of coherent state (Eq. (1.16)) is again valid:

ˆ

a(ω) |{α}i = α(ω) |{α}i , (1.42)

considering that the notation |{α}i indicates a state distributed in the mode α(ω). Or alternatively,

ˆ

aα|αi = α(ω) |αi , (1.43)

where we incorporated the mode properties in the annihilation operator instead of into the state. The only change with respect to the theory presented in Section 1.2.2 regards the commutation rule between the creation and annihilation operator acting on a coherent state:

hα|âα, â†α |αi = h{α}| â(ω), â

†_{(ω) |{α}i = hˆ}

ni. (1.44)

This is a small price to pay because it leaves all the other coherent state relations unchanged. The displacement operator in the distributed mode formalism has the form:

ˆ

D({α}) = eaˆ†α−ˆaα_, (1.45)

that acts on the vacuum generating a coherent state in the mode dened by α(ω). Also the relations regarding the electric eld and the photon number operator remain unchanged.

Summarizing, the distributed mode formalism is useful to take into account many experimental aspects, but, if we are able to generate all the optical states in the same mode, most of the results obtained in Section 1.1 and 1.2 remain valid[1].

1.3 Quantum Measurements

In this section I will link the observables, described from a theoretical point of view in the rst part of this work, to the real measuring devices used in the laboratory during the experiments.

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1.3.1 Photon Number Sensitive Detectors

As I showed in Section 1.2, the photon number distribution of an optical state can give us useful information to understand its nature. Currently, there is a class of detectors, called photon number resolving (PNR), able to detect the exact number of photons in an optical state. These detectors are commonly based on superconducting systems and thus require a complicate cooling apparatus. This technical diculty makes the characterization of an optical state based on the photon number distribution not suitable for many experiments. A more "user-friendly" solution are the so called single photon counting modules (SPCM). They are devices capable to detect, with a given quantum eciency η, the presence or the absence of photons, but not to discern their exact number. Practically, their output is the same (an electrical pulse or a "click") if one or n photons impact on the detector, however large n is, while no clicks are produced if 0 photons arrive on the detector. Due to this fact they are commonly called on-o detectors. This behavior can be mathematically formalized by two POVMs [18](Positive-Operator Valued Measure) ˆΠon = ˆI − |0i h0|and ˆΠof f = ˆI − ˆΠon= |0i h0|,

where ˆI is the identity operator[19].

Using such a detector we can obtain the probability to observe more than one photon in a given optical state. By the denition of POVM, the probability to obtain a click from an on-o detector is Pclick = T r{ ˆΠonρ}ˆ , where ˆρ is the density operator describing the

state we are measuring. Let's consider for example the state ˆρ = (a |0i + b |1i)(a∗_{h0| +}

b∗h1|), that is the superposition between the rst two Fock states described in Section 1.2.1. The probability to observe a click is

Pclick = T r

n

( ˆI − |0i h0|)(a |0i + b |1i)(a∗h0| + b∗h1|)o

=

∞

X

n=0

hn|ρ − |a|ˆ 2|0i h0| − ab∗|0i h1||ni

= |a|2+ |b|2− |a|2 _{= |b|}2_,

(1.46)

that is the probability to observe the single photon component of the state used as example.

It should be emphasized that this type of detector can not be used to measure the pho-ton number properties of an intense light beam, unless they are used in a multiplexed scheme. Since they are designed to detect the small amount of energy carried by a single photon, they have a high-gain amplication stage, which can be easily saturated (and even damaged) by a high-intensity light state.

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1.3.2 Homodyne Detection

As already anticipated in the rst Section, one of the most used techniques to obtain information about a quantum state of light is Homodyne Detection (HD). Using this technique it is possible to directly measure the electric eld quadratures also for the very weak elds of a few-photon state. Unlike the device described in Section 1.3.1, this technique is sensitive to the phase of the optical state, so it is the perfect tool to investigate phase dependent quantum properties like squeezing, or to perform full reconstructions of the density matrix describing a quantum state [20]. In Figure 1.4 a scheme of this measurement apparatus is reported.

Figure 1.4: Schematic representation of the Homodyne Detection (HD) apparatus. The central element, denoted by the abbreviation HDBS, is a balanced beam-splitter, on which the reference beam called Local Oscillator (ˆaL) is mixed with the unknown state

(ˆaS) under analysis. The outputs of this beam-splitter are detected by two photodiodes,

then the dierence between the two photocurrents is amplied and measured via an electronic system.

The state of the mode that we want to analyze, represented by the operator ˆas, is mixed

in a balanced (50:50) beam-splitter with a strong coherent state, usually called Local Oscillator (LO). In this picture the LO beam is represented by the operator ˆaL. The

outputs of the beam-splitter are detected by two photodiodes, the dierence between the two photocurrents is amplied and then measured. It is possible to show that the output of such a detector is proportional to the quadrature of the unknown quantum state of the signal mode, measured at the relative phase between the local oscillator

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and the signal. Due to the central role played by the balanced beam splitter, I remind here the laws that describe its behaviour:

ˆ

d1 = r ˆaS+ t ˆaL

ˆ

d2 = t ˆaS+ r ˆaL.

(1.47) For a 50:50 beam splitter the reection and transmission coecients can be written in the following form:

r = √i

2 t =

1 √

2, (1.48)

so the operator describing the Homodyne measure is: ˆ H− ≈ ˆn2− ˆn1 = ˆd † 2dˆ2− ˆd † 1dˆ1 = ihâ†_SâL− â † LâS i . (1.49)

If we calculate the expectation value of this operator, considering a strong coherent state for the LO (|αLi) and a generic state in the signal mode (|ΨSi), we can see the

link with the quadratures operator acting on the signal mode: hαL, ΨS| ˆH−|ΨS, αLi = i hαL, ΨS| h ˆ a†_SâL− â † LâS i |ΨS, αLi = 2√nLhΨS| hâ† Se i(θL+π₂)_{+ ˆ}_a Se−i(θL+ π 2) 2 i |ΨSi = 2√nLh ˆXSi, (1.50)

where we can recognize the denition of the quadrature operator acting on the signal ( ˆXS). The phase of the quadrature measurement (θM), dened in Equation 1.6, is

strongly linked to the LO phase (θL), but, for a correct determination of this parameter,

we have to take into account also the phase of the state on wich we are performing the measure. The quadrature value obtained with an Honodyne measurement is amplied by a factor (√nL) proportional to the intensity (mean photon number) of the LO beam.

As already mentioned, the concept of mode is of a great importance in the homodyne measurement. Thus, before going further with HD description, let us switch to the distributed mode formalism considering that the local oscillator is a strong coherent state |αLi, occupying the mode α(ω) = |α(ω)|eiθL, and |ψSi is a generic quantum state

distributed in the mode β(ω) = |β(ω)|eiθS, that in principle can be dierent from the

LO one7_{. If we calculate the expectation value of the HD operator using this formalism,}

7_{As in Section 1.2 I consider only the spectral mode of the optical states. A complete description}

of Homodyne measurement requires to take into account the whole mode function of both the LO and the unknown state |ΨSi.

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we have: h ˆH−i ≈ Z dω hψS| hαL| i β∗(ω)â†_Sα(ω)âL− α∗(ω)â † Lβ(ω)âS |αLi |ψSi = Z dω|αL(ω)||βS(ω)| hψS| â † Se i(θL−θS+π₂)_{+ ˆ}_a Se−i(θL−θS+ π 2)|ψ_Si = 2√nL Z dω|˜αL(ω)||βS(ω)| · · hψS| ˆXScos(θL− θS + π 2) + ˆYSsin(θL− θS+ π 2) |ψSi . (1.51)

In the last line I used the denition α(ω) =√nLα(ω)˜ , where α(ω) is the coherent state

mode prole dened in Equation (1.41), while ˜α(ω) is the prole normalized to 1. From Equation (1.51) the central role played by the local oscillator emerges. First of all, as already seen in Equation 1.50, the term √nL acts as an amplication factor. It is the

square root of the local oscillator mean photon number, that, due to the large intensity of this beam, can amplify the small signal coming from the unknown quantum state |ψSi

above the electronic noise of the measurement apparatus. The term R dω|˜αL(ω)||βS(ω)|

has a deep meaning too. It quanties our ability to match the mode of the state that we want to measure to the mode of the LO, dening the so called mode-matching eciency, ηmm = (

R

dω|˜αL(ω)||βS(ω)|)2. A good mode matching is fundamental to

perform homodyne detection, indeed, having xed the mode of the local oscillator, if the state that we want to characterize is not in the same mode, this integral may drop to zero, despite the eect of the amplication. Practically, the local oscillator acts as a lter, dening the mode over which the homodyne measurement is performed. From Equation 1.51 we have also a precise denition of the measurement phase θM

introduced in Equation (1.6): apart from the constant term π

2, this parameter is xed

by the relative phase between the LO and the state we are measuring, θM = θL−θS+π₂.

We can redene the homodyne operator as: ˆ H−(θM) ≈ 2 √ nL √ ηmm ˆXScos(θL− θS+ π 2) + ˆYSsin(θL− θS+ π 2) = 2√nLXˆSα˜(θM), (1.52)

where the operator ˆXα˜

S(θM) is the distributed mode version of the operator dened

in Equation (1.6), that describes a quadrature measurement acting on the unknown quantum state |ψSi, performed in the mode ˜α, at the measurement phase θM = θL−

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Losses

The approximation symbol used in the Equation (1.52) reminds us that another step has to be done to keep into consideration all the technical details of this measurement. In the previous discussion an implicit assumption has been done: the two photodiodes are considered as perfect. In the real world, just a portion of the impinging photons are converted into electrons detectable via the acquisition system. This portion is usually quantied via the quantum eciency ηph parameter, that is usually given by

the photodiodes manufacturers. A model that describes this phenomena makes use of an unbalanced beam splitter as shown in Figure 1.5.

Figure 1.5: Graphical representation of the loss beam-splitter model used to take into account the non-unitary detection eciency of a photodiode (ηph). The signal under

test (ˆaS) is mixed with a portion of vacuum (ˆν0) proportional to ηph.

This model gives us the possibility to keep considering ideal detectors at the cost of placing, in front of them, a beam-splitter that mixes the state that we want to measure (ˆaS) with a portion of vacuum (ˆν0) proportional to the quantum eciency of

the detector. Introducing this model, the homodyne apparatus appears as shown in Figure 1.6.

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Figure 1.6: Same apparatus of Figure 1.4 in which we implemented the loss beam-splitter model to consider the eect of non-perfect photodiodes.

We can now calculate the operator describing the homodyne measurement using the relations: ˆ d1 = r ηph 2 (iâ β S+ â α L) + ip1 − ηphνˆ0 ˆ d2 = r ηph 2 (â β S+ iâ α L) + ip1 − ηphνˆ0, (1.53) where âβ

S is the operator representing the signal state in the mode β(ω), ˆa α

L represents

the local oscillator beam in the mode α(ω) and ˆν0 is the operator acting on the vacuum

mode added to consider the losses, ˆ H−(ηph, ηmm, θM) = ˆd † 2dˆ2− ˆd † 1dˆ1 = iηph ˆ a†β_S âα_L− â†α_L âβ_S+ + i r ηph(1 − ηph) 2 (i + 1)(â†β_S − â†α_L )ˆν0− (i − 1)(âβS− â α L)ˆν † 0 . (1.54)

The expectation value of this operator, calculated on the same signal and local oscil-lator state used in the ideal description made before, claries the eect of non-perfect photodiodes. Unlike the approximate version, in this case we also have to consider the

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S(θM)has the same meaning explained for Equation (1.52), incorporating also

the eect of non-ideal photodiodes. In this calculation the second term of Equation (1.54) drops to zero due to the relations h0| ˆν0|0i = h0| ˆν0†|0i = 0. From Equation

(1.55) we see that non-ideal photodiodes acts in the same way of a non-perfect mode-matching, reducing both the signal and the LO amplitude. Starting from this point we can operatively summarize all the unavoidable losses in the experiment with an overall detection eciency ηdet. This factor will take into account, along with the mode

matching and the photodiode eciency, also the electronic noise due to an imperfect amplication and subtraction of the electronic signals (ηel)[21] and optical losses caused

by the unavoidable imperfections of the optical devices (ηop)

ηdet= ηmm· ηph· ηel· ηop. (1.56)

Going further in the HD description we can calculate the variance of the ˆH− operator

to show that it is related to the electric eld quadrature variance of the unknown state. For the sake of simplicity let's perform this calculation considering, as the unknown state, a Fock state in the mode β(ω), |ψSi = |nSi

∆ ˆH− 2 = h0, nS, αL| ˆH 2 −|αL, nS, 0i − h0, nS, αL| ˆH−|αL, nS, 0i 2 = 4nLη2phhnS| 1 2(ηmmnˆS + 1 2) |nSi + nLηph(1 − ηph). (1.57) The rst term of Equation (1.57) is the variance of the electric eld measured on a Fock state (see Eq. (1.12)), incorporating also the eect of a non perfect mode-matching. In the ideal case of perfect photodiodes and perfect mode-matching, the variance of the HD signal is exactly the variance of the electric eld carried by the state we are measuring. If instead, we totally fail to mode-match the local oscillator to the signal mode (ηmm = 0), we have: ∆ ˆH− 2 shot = nLηph, (1.58)

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that is the variance of the vacuum or the shot noise level of the detector. As we al-ready saw in the ideal version of the HD described at the beginning of this section, the local oscillator can amplify also the lowest optical signal above the electric noise of the detector. This relation is often used to identify the linear region of an homodyne detector. The unbalancing of the homodyne beam-splitter, saturation of the photodi-odes or dierences between them can cause a distortion of the signal and so a deviation from the linearity predicted by Equation (1.58).

As last step in the description of Homodyne Detection technique I want to introduce a graphical tool useful to represent the results of a HD measurement, usually called phasor diagram. It consists of a 2D chart in which the two axes represent the real and imaginary components of the electric eld ( ˆX and ˆY ) carried by the state. We already dened these quantities in Section 1.1 starting from the general denition of the quadrature operator

ˆ XθM = ˆ a e−iθM _{+ ˆ}_a†_eiθM 2 = aˆ †_{+ ˆ}_a 2 cos(θM) + i ˆ a†− ˆa 2 sin(θM) = ˆX cos(θM) + ˆY sin(θM). (1.59)

For example, an Homodyne measurement, performed at a local oscillator phase of θLO

on a single mode coherent state of amplitude |α| and phase θα will be represented by

the phasor diagram in Figure 1.7.

Figure 1.7: Phasor diagram representing an homodyne measurement performed on a coherent state.

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state, while the circle centered at its extremity takes in account the unavoidable un-certainty of each quadrature measurement due to the Heisenberg unun-certainty principle. By changing the measurement angle, the expectation value of the Homodyne measure-ment changes according to Equation (1.19) and it is represented by the projection of the red line on the X axes in the phasor diagram.

1.4 State Tomography

In this chapter we saw that dierent states have dierent properties (for example, the photon number and electric eld distribution) and also that these properties can be collected in the density operator ˆρ, which fully describes a quantum system. In the next two chapters, I will show dierent techniques to manipulate optical quantum states. I will explore a method to emulate strong nonlinear eects at the single photon level, or how to generate entanglement between macroscopic states. To get the proper conclu-sions from these experiments a complete characterization of the nal states is needed. A procedure that allows one to know all the information about a system, without us-ing a priori knowledge about it, is usually called quantum state tomography. In many cases, this procedure allows us to know the density matrix elements of the generated state expressed in a given basis8_{. As already seen in the previous section, homodyne}

detection is a powerful tool to measure the amplitude of the electric eld carried by a quantum state, for dierent phases. In this section I will explain the procedure used during this work to reconstruct the density operator describing a given state from a set of homodyne measurements.

Maximum Likelihood Algorithm

This technique is based on the maximization of a functional widely used in mathemat-ical statistic, called Likelihood (L). L represents the expected probability of observing a given set of values {yi} as a consequence of a measurement on a system described

by the density operator ˆρ. Therefore, this procedure can be splitted into two distinct steps: the rst is the collection of a proper data set, while the second is the maxi-mization of L over all the possible ˆρ. The set {yi} must be carefully selected because

not all the observables give the complete information about the system. For example,

8_{To analyze the results of our experiments, we always express the reconstructed density matrices}

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measuring only the position of a particle in a harmonic potential is not sucient to determine its wave function, it will also be necessary to measure its momentum. To fully characterize the states I will discuss in the rest of this work, I will use a set of quadrature operators ˆXi(θM), each measured for dierent values of θM.

For the second step, we make use of the work of Hradil et al. [22]. They demon-strated, using variational methods, that the maximum value of the likelihood can be found by solving the non linear system:

ˆ R( ˆρ, y) ˆρ ˆR( ˆρ, y) = ˆρ, (1.60) where ˆ R( ˆρ, y) =X i fi |yii hyi| hyi| ˆρ |yii , (1.61)

and fi is the frequency of the measured value yi.

Assuming the initial condition ˆρ(0) _{= N ˆ}_I, where N is a normalization factor, the

system (1.60) can be iteratively solved according to the equation ˆ

R( ˆρ(n), y) ˆρ(n)R( ˆˆ ρ(n), y) = ˆρ(n+1). (1.62) Before entering into the details of this calculation, it is useful to better specify our data set:

{yi} = {xi(θj), θj}, (1.63)

that takes into account the fact that the same value of xi can be measured for dierent

choices of θ. The POVM operators describing the measurements performed to obtain the above data set have the form:

|yii hyi| = |xi(θj), θji hxi(θj), θj| . (1.64)

To implement the abstract algorithm of Equation (1.62) it is necessary to project it on a properly chosen basis and give a specic form to the operators ˆR( ˆρ(n)_{, y)} _{and ˆρ.}

Describing the number states I said that, in many cases, this is the simplest base to perform calculation. This is one of those cases. Indeed, we have:

ρ(n+1)_m,n = hm| ˆR( ˆρ(n), y) ˆρ(n)R( ˆˆ ρ(n), y) |ni =X k,l hm| ˆR( ˆρ(n), y) |ki ρ(n)_k,l hl| ˆR( ˆρ(n), y) |ni =X k,l Rm,k( ˆρ(n), y)ρ (n) k,lRl,n( ˆρ (n)_{, y),} (1.65)

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where ρ(n+1)

m,n is the matrix element of the density operator at the iteration n + 1, the

matrix element Rm,k( ˆρ(n), y) has now the form:

i(θj), θj)the probability to observe the quadrature xi at the angle θj

cal-culated from the density matrix at the nth iteration P r(n)(xi(θj), θj) = X t,s hxi(θj), θj|tiρ (n) t,shs|xi(θj), θji. (1.67)

Using these ingredients, and reminding that the projection of the quadrature eigen-states on the number eigen-states is:

hn|xi(θj), θji = einθ 2 π 1₄H_n( √ 2x) √ 2n_n! e −x2 , (1.68)

solving the system (1.62) is just a matter of computation, apart from an unavoidable assumption. To calculate the series (1.65), (1.66) and (1.67) they have to be truncated to a nite number of terms. This means that a proper assumption must be introduced to limit the Fock space9_{. Even if it is possible to do this without using any hypothesis}

on the state under analysis, in many experiments a certain amount of a priori knowl-edge is always present. The intensity of the measured state, some phase relation or the number of involved modes can help in this operation, thus simplifying the calculations. Another interesting feature of the maximum likelihood algorithm is the possibility to include the eects of non-unitary detection eciency adopting the model of the lossy beam-splitter introduced for the homodyne detection. We have to reconsider the ingre-dients of the algorithm as transformed by an η transmissivity beam-splitter [23]. This adjustment allows the reconstruction of what has been really experimentally generated without considering the deterioration due to a non-perfect observation.

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Chapter 2 Single-Mode State Manipulation

In the previous chapter, I described the features of the main quantum optics operators. Among these operators the most fundamental two are, of course, ˆa and ˆa†_{. They}

have been analyzed in detail, considering also their distributed mode version to give a description more suitable for their practical realization. In this chapter I will discuss the techniques used during my PhD to experimentally implement the annihilation and the creation operators and superpositions of those on the same mode. Both of them share a fundamental aspect: they are based on a probabilistic approach. In contrast to an on-demand realization in which the wanted operation is realized at a specic moment, decided by the experimenter, in the probabilistic implementation the operation is randomly applied to the input state. Due to the ignorance about the application or not of the operation, this approach seems more suitable to produce, not the wanted operation ( Ô), but a mixture between it and the identity operator (Î), which is the mathematical way to say that "nothing happened". To overcome this problem, in the probabilistic implementation we make use of a second mode, usually called ancillary mode. Making a properly chosen measurement on the ancillary mode, it is possible to herald1 _{the successful application of the operator ˆ}_O_{, erasing the contribution of Î. So}

we can represent the tools used to experimentally realize ˆa and ˆa† _{as a machine with}

two inputs and two outputs. Two of them, labeled with 1, will be used to describe the signal before and after the operation, while the other two (labeled with a 2) will describe the ancillary mode. We can mathematically formalize this idea using two

1_{We usually refer to this type of approach as heralded (or conditional) implementation or}

measure-ment induced operation.

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unitary operators with the form ˆ U{sub,add}= eiγ ˆJ (â1,â2,â † 1,â † 2), (2.1) where ˆJ (â1, â2, â † 1, â †

2)is the hermitian operator that generates the wanted

transforma-tion. For the annihilation case we should use ˆ Jsub(â1, â2, â † 1, â † 2) = â1aˆ † 2+ â † 1â2, (2.2)

while in the addition case ˆ Jadd(â1, â2, â † 1, â † 2) = â † 1â † 2+ â1â2. (2.3)

The operator (2.1) can be the starting point to implement the creation and the annihi-lation operators only in the case of a very low success probability. Indeed (2.1) leads to the desired operator only at the rst order of approximation respect to the parameter γ, that, at this abstract level of description, can be only linked to the strength of the cou-pling between the modes 1 and 2. As I said, this probabilistic approach can not works if the ancillary mode is not properly detected2_{. So we have to consider as fundamental}

part of the machine used to realized a desired operation also the heralding system that measures the mode 2. To better understand this method let's focus on the annihilation operator case, considering that at this general level the same considerations stand for the creation operator. Taking γ ≈ 0 we can write:

ˆ

Usub|ψi₁|0i₂ = N {h ˆI + iγ

ˆ a1aˆ†2+ â † 1â2 + O(γ2) i |ψi₁|0i₂} ≈ N {|ψi₁|0i₂+ iγâ1|ψi1|1i2},

(2.4) that represents, for the signal mode, the superposition between the initial state and the same state after the application of the annihilation operator, with probability γ2_.

Looking at the whole state of Equation (2.4), composed by the signal and the ancillary modes, we can see that, when the ˆa1 operator acts on the input state, there is a single

photon in the mode 2, while for the unmodied part this mode remains in the vacuum state. So the detection of a single photon in the mode 2 will let us know when the operation has been performed. If we place an SPCM along the ancilla path, we can remove the unmodied component by observing the signal mode only in coincidence with a click from the detector3_,

N {|ψi₁|0i₂+ iγˆa1|ψi1|1i2}

heralding

−−−−−→

ˆ

I2−|0i2h0|

iγˆa1|ψi1. (2.5)

2_{This can be easily understood thinking to the fact that the operator 2.1 is unitary, while the}

operators that we want to implement (ˆa and ˆa†₎_{are not.} 3_{See Section 1.3.1 for more details.}

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It doesn't matter how small the success probability is, every time the operation is performed it is announced by the detection of a single photon in the ancillary mode. Of course, neglecting the higher order terms in the expansion of ˆUsub (Eq. (2.4)) leads

to unavoidable errors. The faithful implementation of the annihilation operator using this technique requires that the probability (γ4_{) to perform the operation (ˆa}

1)2|ψi₁

is negligible respect to γ2_{. Thus, it is clear that the γ parameter must be chosen as}

the best compromise between this request and the necessity to keep experimentally acceptable success probabilities.

2.1 Experimental Realization of Single Photon

Subtraction

The single photon subtraction operation has been fundamental for the realization of important experiments during the years, like Schrodinger's cat state generation [24], enhanced quantum metrology [25] and fundamental tests [26][5]. The idea at the base of its implementation is simple: the subtraction of a single photon from a traveling optical state can be seen as a controlled loss. In this scenario, the ˆa operator can be realized using a low-reectivity beam-splitter. Indeed, the operator that obeys the relations (1.47) has exactly the form that we are looking for

ˆ Usub= ÛBS = eiγ(â1â † 2+â † 1â2). (2.6)

It is possible to better see it if we look at what happens injecting a single photon in an arm of a beam-splitter as described by the relation used in the previous chapter,

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compared to the application of the operator ˆUBS on the same state:

Chapter 1 Description |1i₁|0i₂ = ˆa†₁|0i₁|0i₂ −−−−−−−−−→BS

Equations (1.47)

tˆa†₁+ rˆa†₂|0i₁|0i₂

= t |1i₁|0i₂+ r |0i₁|1i₂.

Application of ÛBS ˆ UBS|1i1|0i2 = ÛBSâ † 1|0i1|0i2 = ÛBSâ † 1Uˆ † BSUˆBS|0i₁|0i₂

=ˆa†₁cos(γ) + iˆa†₂sin(γ)|0i₁|0i₂ = cos(γ) |1i₁|0i₂+ i sin(γ) |0i₁|1i₂,

where I used the property of ˆUBS to be unitary ( ˆUBSUˆ †

BS = 1), and the relation

ˆ

UBS|0i₁|0i₂ = |0i₁|0i₂, that means that a two mode vacuum state remains unchanged

after passing through a beam-splitter. From these relations we can see that, if we consider t = cos(γ) and r = i sin(γ), the operator ˆUBS actually describes the behavior

of a beam-splitter. They are also telling us that, if we look at the signal mode after the beam-splitter, in most cases, with the probability cos2_(γ)_{, the injected state}

re-mains unchanged4_{, while, with the probability sin}2_(γ)_{, a photon is removed from the}

signal mode with the generation of a photon in the ancillary one. At this point, the strong link between the physical meaning of the parameter γ and the strategy used to practically implement the desired operation (ˆa) is more clear. In the experimental implementation of the annihilation operator using a low-reectivity beam-splitter we have γ = arccos(t), that shows the link between the abstract γ and the more practical reectivity of the beam-splitter (t). This implementation of the ˆa operator has been used in many interesting works until now [2]. In Figure 2.1 a schematic picture of the setup used to realize this idea is reported.

In this representation we used the two spatial modes of the beam-splitter, one for the signal and the other for the ancilla, but, from the experimental point of view, it could not be the best choice. Indeed it implies that the two optical states travel along dif-ferent paths, suering from dierent losses and phase uctuations. In some cases, a better way to realize the ˆa operator is making use of the polarization degrees of freedom

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Figure 2.1: Schematic view of the heralded single photon subtraction scheme.

of the light states, with the help of a few polarization sensitive devices. In this type of implementation, the role of the spatial modes is now played by the two orthogonal polarization components of the traveling wave, that we will call horizontal and vertical. For example, we could consider the signal mode as the portion of an optical beam with horizontal polarization, while the vertically polarized part as the ancilla mode. To obtain the same eects seen using the spatial degrees of freedom, we need an optical device that mixes the two modes, like the beam-splitter. Its analog for the polarization encoding is the half-wave-plate (HWP). This optical device is a foil of a birefringent material cut with a proper thickness such that a delay of π is inferred between the extraordinary (ordinary) and ordinary (extraordinary) polarization components, for a xed wavelength. This means that a linear polarization going through the plate will be rotated of an angle γ when the ordinary axis of the birefringent crystal is rotated of an angle γ

2 with respect to the input polarization direction5. Such a device is the

analogous of a beam-splitter, indeed its input/output relations have the form: ˆ aout_H ˆ aout_V ! = cHH cHV cV H cV V ! ˆ ain_H ˆ ain_V ! , (2.7)

that, considering cHH = cV V = cos(γ) and cHV = cV H = i sin(γ), with the particle

number conservation law ˆnin

H + ˆninV = ˆnoutH + ˆnoutV , are formally identical to the

beam-splitter ones. In this case, the small success probability, required to consider valid the relation (2.4), can be obtained with small rotations of the signal mode polarization with respect to the ordinary axis of the HWP [27]. For example, by setting this angle

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to 5 degrees, from the Malus' law we nd a success probability of ≈ 3%, which is the one typically used for most of our experimental implementations. In these conditions, a photon from the polarization mode of the signal is transferred to the one of the ancilla. However, both the modes share the same spatio-temporal prole, experiencing the same drifts or vibrations of any optical element, enabling for a particularly stable and compensation free implementation of ˆa. The last step, also required in the polarization based realization of ˆa, is the detection of the ancillary photon. To do this, we need to spatially separate the two polarization modes. This can be achieved by using another birefringent device, a polarizing beam-splitter (PBS). This device can be considered as a normal beam-splitter despite the fact that the transmission and reection coecients depend on the polarization of the incoming beam. Using a PBS it is possible to totally transmit a polarization component, i.e. the horizontal polarization with respect to the PBS axes, while the other one is totally reected. In this way, the two modes travel separately only at the end of the experimental scheme, the signal towards the char-acterization apparatus and the ancilla toward the heralding detector. As remarked in the previous chapter, to allow the treatment of the annihilation operator in the single mode picture we have to perform the operation in the same mode of the signal. To achieve this condition, the SPCM used to herald the operation has been coupled to the target beam by means of a single mode ber. The spatial propagation mode of the ber has thus been matched to the spatial mode of the target state by using a system composed of two lenses, resulting in an eciency of ≈ 80% of ber-coupled photons. In Figure 2.2 is reported a schematic view of the setup just described.

Figure 2.2: Schematic view of the heralded single photon subtraction scheme based on the polarization degrees of freedom of the light.

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For all the experiments described in this work I used a mode-locked Ti:sapphire laser emitting 1.5 ps long pulses at 786 nm, with a repetition rate of 80 MHz. Consider-ing also that the SPCM used for the heraldConsider-ing has a detection eciency of ≈ 60%, we can expect a success rate for the annihilation operation of ≈ LaserRepetionRate · SubtractionProbability·FiberCouplingEciency·SPCMEciency ≈ 1.1MHz , if at list one photon per pulse is present at the input.

2.2 Experimental Realization of Single Photon

Addition

Figure 2.3: Schematic view of the heralded single photon addition scheme.

The generation of single photons, that in other words is the application of the ˆa†

operator to the vacuum state, is an "hot topic" in the quantum experimental world. As already anticipated during the description of the number states, a well known process to achieve this purpose is Parametric Down Conversion (PDC) [2]. Thanks to this process, it is possible to add a single photon in an arbitrary mode, both to the vacuum state, and so we will talk about Spontaneous Parametric Down Conversion, or to an arbitrary state (Stimulated Parametric Down Conversion). The core of each PDC-based implementation of the ˆa† _{operator is a nonlinear optical medium. It should}

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have a second order nonlinear coecient (χ(2)_{) dierent from zero. If we inject in}

such a material an intense laser beam, called pump, there is a non-zero probability that a photon of this beam is annihilated by the interaction with the atoms of the medium. This interaction is not resonant with any atomic level, so the pump photon is annihilated by exciting a very short-lived virtual atomic state with the simultaneous emission of two single photons at lower energy, usually called signal and idler. One of them, usually the signal one, will be the photon added to the target state, while the other will be detected, heralding the success of the operation, as shown in the schematic setup of Figure 2.3. For all the experiments described in this thesis we used a 3-mm long bulk β−barium borate (BBO) crystal as nonlinear medium. For a better understanding of the PDC process let's start from the light-matter interaction hamiltonian ˆ H = Z V ˆ P (r, t) · ˆE(r, t)dr, (2.8)

where ˆP (r, t) describes the polarization of the medium and V is the crystal volume. The l-th polarization component of this operator can be written as a series of electric eld operators as ˆ Pl(r, t) = χ (1) lm(r) Êm(r, t) + χ (2) lmn(r) Êm(r, t) Ên(r, t) + ... (2.9)

where the indices l, m, n run over the two polarization components and the summation on the repeated indices is assumed. χ(1)

lm is the lm component of the linear susceptibility

tensor while χ(2)

lmn is its second order nonlinear term. The ˆE(r, t) operator of Equation

(2.8) is the electric eld representing the pump for the nonlinear process, and has the form ˆ El(r, t) = Z ˆ al(k, ω)e−i(k·r−ωt)+ â † l(k, ω)e i(k·r−ωt)dkdω = Ê(+) l (r, t) + Ê (−) l (r, t). (2.10) Since we are interested in the rst order nonlinear process, we will consider only the rst nonlinear term in the polarization decomposition. Among all the possible nonlinear processes we want to describe the one in which a pump photon is annihilated with the consequent generation of two lower-energy photons. So, inserting the right terms of the expressions (2.9) and (2.10) in the Equation (2.8), we nd the interaction hamiltonian

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ˆ HP DC = Z V χ(2)_lmn(r) Ê_l(+)(rp, tp) Êm(−)(rs, ts) Ên(−)(ri, ti)dr = Z V χ(2)_lmn(r)âl(kp, ωp)âm† (ks, ωs)â†n(ki, ωi)· · e−i[(kp−ks−ki)·r−(ωp−ωs−ωi)t]dk pdωpdksdωsdkidωidr, (2.11)

The indices p, s and i indicate that the relative operators act on the pump, signal and idler modes respectively. This hamiltonian governs the temporal evolution of the three involved modes according to the Schr¨odinger equation

i~d

dt|Ψ(t)i = ˆHP DC(t) |Ψ(t)i . (2.12)

Considering that this evolution can be described also in terms of the unitary operator ˆ

UP DC(t) according to the equation

|Ψ(t)i = ˆUP DC(t) |Ψ(0)i , (2.13)

we can put the Equation (2.13) in the (2.12) and solving for ˆUP DC(t). The solution is

ˆ

UP DC(t) = e−i Rt

−∞HˆP DC(t0)dt0_, (2.14)

that, apart from the integral required to dene the shape of the mode, has the right form to describe the creation operator, according to the Equation (2.1)6_{. As we did}

for the annihilation operator we have to expand the operator ˆUP DC(t) as a series of

powers, but in this case we will do it respect to χ(2)

lmn. These parameters are of the

order 10−11_{÷ 10}−8 _{so we can stop the expantion at the rst order of approximation.}

This assertion is conrmed by the experimental data. Considering the same apparatus parameters used at the end of the previous section, whit 100 mW of pump power at 393 nm of wavelength, we have Pp hνp = 0.1 W 6.6 · 10−34_{J · s · 7.5 · 10}14_s−1 ≈ 2 · 10 17_s−1 _(2.15)

injected photons per second in the nonlinear crystal. An averaged value for the idler count rate measured with our setup is ≈ 2000 cps. So we can estimate the success single photon addition probability as

2000 s−1

2 · 1017_s−1 ≈ 10 −14

, (2.16)

6_{To clarify the notation, the signal mode, indicated with the label 1 in Equation (2.1), is here}

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where we are not considering the optical losses along the signal path. This number means that we generate, on average, a signal/idler photon pair every 1014pump photons

that traveled in the crystal. This data has to be compared with the probability to perform a double addition (ˆa†₎2_{, that is of the order 10}−28 _{for each pump photon. So}

we can use the approximated version ˆ UP DC(t) ≈ ˆI − i Z t −∞ ˆ HP DC(t0)dt0, (2.17)

instead of the Equation (2.14). To go further in the study of the Parametric Down Conversion process we have to dene the initial state of the three involved modes, represented in the previous equations by |Ψ(0)i. Generally the pump beam is a strong laser beam, so it can be considered as a coherent state with a very large mean photon number (|{α}ip) occupying the mode αl(kp, ωp). For sake of simplicity we can start

considering the signal and the idler in the vacuum state before the nonlinear crystal. So the initial state before the PDC is

|Ψ(0)i = |{α}i_p|0i_s|0i_i. (2.18)

We can now study the so-called phase-matching conditions that have to be satised to perform an ecient PDC. Applying the operator (2.17) to the state (2.18) we can obtain a preliminary expression of the PDC output state

· ˆa†_m(ks, ωs)ˆa†n(ki, ωi) |αi_p|0i_s|0i_idkpdωpdksdωsdkidωidt0dr,

(2.19)

where ∆k = kp−ks−ki and ∆ω = ωp−ωs−ωi. Considering that we are not interested

in the description of the signal state during the interaction inside the crystal, and also that ˆHP DC(t0)is zero before and after the interaction, we can extend the time integral

to +∞, Z +∞

−∞

ei∆ωt0dt0 = δ(ωp− ωs− ωi). (2.20)

Equation (2.20) is the rst phase matching condition. It is the energy conservation law that denes the relation between the pump and the signal/idler photons frequency.

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Regarding the spatial integral, considering the χ(2)

lmn susceptibility constant over the

crystal volume is a reasonable assumption. So it can be solved as follows K(∆k) = Z V e−i∆k·rdr = Z +Lx₂ −Lx 2 e−i∆kxxdx Z +Ly₂ −Ly 2 e−i∆kyydy Z +Lz₂ −Lz 2 e−i∆kzzdz = 8sin( ∆kxLx 2 ) ∆kx sin(∆kyLy 2 ) ∆ky sin(∆kzLz 2 ) ∆kz . (2.21)

This term denes the spatial bandwidth of the crystal. Indeed, it can be considered as a sort of spatial lter that allows the generation of the signal and idler photons only for specic combinations of the wave vectors. Equation (2.21) has a maximum for

kp− ks− ki = 0, (2.22)

that is the second phase matching condition and tells us that if we observe an idler photon along the direction ki, given the pump direction kp, we have the maximum

probability to nd the signal photon with ks= kp−ki. The width of the ksdistribution

is smaller as the crystal is longer, becoming the usual momentum conservation law in the limit of innite crystals. We can now rewrite the PDC output state in the simplied form

|Ψ(t)i = |Ψ(0)i − Z

φlmn(ks, ωs, ki, ωi)ˆa†m(ks, ωs)ˆa†n(ki, ωi) |αi_p|0i_s|0i_idksdωsdkidωi

= |Ψ(0)i − Z

φlmn(ks, ωs, ki, ωi)dksdωsdkidωi|αi_p|1ks,ωsis|1ki,ωiii,

(2.23) where we dened the shape of the output modes of the spontaneous parametric down conversion process as φlmn(ks, ωs, ki, ωi) = iχ (2) lmn Z αl(kp, ωs+ ωi)K(∆k)dkp. (2.24)

We have to notice that this prole is the convolution between the pump prole and the crystal band. We have to be careful in the interpretation of φlmn(ks, ωs, ki, ωi) as

mode prole of the signal and idler photons. They are a pair of quantum correlated (entangled) photons, so their individual properties are not well dened until one of them is detected, only at that time the mode distribution of the other one will be well dened7_.